External boundaries of pole localization region formulation for transfer function with interval-given parameters

In this paper the approach for external boundary of pole localization region formulation for transfer function with interval-given parameters is proposed. The boundary is formulated as analytic piecewise function of characteristic polynomial parameters of the given transfer function.

Analytic formulation of external boundary of poles localization region allows to reduce computations since existing methods require iterative numeric calculations of characteristic equation roots with fixed step size for edges mapping or full interval root locus mapping as well. Formulated boundary allows to clearly describe system behavior and calculate variation ranges of performance indexes. In addition, piecewise function that constrains gives new opportunities for parametric controller synthesis for systems introduced by transfer functions with interval-given parameters.

The results can find its practical application in aerospace engineering problems of mathematical analysis and synthesis for highly-precise systems of self-direction missiles.

In the research the boundary formulation is performed for third order transfer function. Transfer function order was chosen due to the fact that many physical systems and objects can be described mathematically with the third order transfer function, e.g. model of missile target-seeking head with gyro stabilized drive is described with this model.

The research was performed on the basis of the following step sequence: firstly, analytical solving of cubic equation applying Cardano’s formula; secondly, interval root locus edges functions obtaining, next external vertexes set obtaining and, finally, external border formulation and plotting.